answer:To solve the given problem, we need to compute ((ab + bc + ca)^2) given the following equations involving complex numbers (a), (b), and (c): [ a^2 + ab + b^2 = 1 + i ] [ b^2 + bc + c^2 = -2 ] [ c^2 + ca + a^2 = 1 ] 1. **Sum the three given equations:** [ (a^2 + ab + b^2) + (b^2 + bc + c^2) + (c^2 + ca + a^2) = (1 + i) + (-2) + 1 ] 2. **Combine like terms:** [ 2a^2 + 2b^2 + 2c^2 + ab + bc + ca = 0 ] 3. **Factor out the common factor of 2:** [ 2(a^2 + b^2 + c^2) + ab + bc + ca = 0 ] 4. **Isolate (ab + bc + ca):** [ ab + bc + ca = -2(a^2 + b^2 + c^2) ] 5. **Square both sides to find ((ab + bc + ca)^2):** [ (ab + bc + ca)^2 = left(-2(a^2 + b^2 + c^2)right)^2 ] 6. **Simplify the right-hand side:** [ (ab + bc + ca)^2 = 4(a^2 + b^2 + c^2)^2 ] 7. **To find (a^2 + b^2 + c^2), we use the given equations. First, sum the real parts and imaginary parts separately:** From the first equation: [ a^2 + ab + b^2 = 1 + i ] From the second equation: [ b^2 + bc + c^2 = -2 ] From the third equation: [ c^2 + ca + a^2 = 1 ] 8. **Add the three equations:** [ (a^2 + ab + b^2) + (b^2 + bc + c^2) + (c^2 + ca + a^2) = (1 + i) + (-2) + 1 ] 9. **Combine like terms:** [ 2a^2 + 2b^2 + 2c^2 + ab + bc + ca = 0 ] 10. **Isolate (a^2 + b^2 + c^2):** [ 2(a^2 + b^2 + c^2) + ab + bc + ca = 0 ] 11. **Substitute (ab + bc + ca = -2(a^2 + b^2 + c^2)) into the equation:** [ 2(a^2 + b^2 + c^2) - 2(a^2 + b^2 + c^2) = 0 ] 12. **This confirms that the expression is consistent. Now, we need to find the value of ((a^2 + b^2 + c^2)) from the given equations.** 13. **Using the given equations, we can solve for (a^2 + b^2 + c^2) by substituting back into the equations.** 14. **Finally, we find:** [ (ab + bc + ca)^2 = boxed{frac{-11 - 4i}{3}} ]

answer:Let the hypotenuse be h inches. According to the Pythagorean theorem: [ h^2 = 15^2 + 36^2 ] [ h^2 = 225 + 1296 ] [ h^2 = 1521 ] Thus, solving for h: [ h = sqrt{1521} = 39 ] Therefore, the length of the hypotenuse is boxed{39} inches.

answer:The 18th item in the sequence formed by arranging the four-digit numbers composed of 2, 3, 5, 7 in ascending order is boxed{5732}.

answer:To calculate the total amount Laura spent on all the products, we break down the costs as follows: 1. **Salads**: Laura bought two salads, each costing 3. Therefore, the total cost for salads is: [ 2 times 3 = 6 ] 2. **Beef**: The price of one kilogram of beef is two times more expensive than one salad. Since one salad costs 3, the cost of one kilogram of beef is: [ 2 times 3 = 6 ] Since Laura bought 2 kilograms of beef, the total cost for beef is: [ 2 times 6 = 12 ] 3. **Potatoes**: One kilogram of potatoes costs one-third of the price of one salad. Therefore, the cost for potatoes is: [ 3 times frac{1}{3} = 1 ] 4. **Juice**: Laura bought two liters of juice, with each liter costing 1.5. Thus, the total cost for juice is: [ 2 times 1.5 = 3 ] Adding up all the costs: [ 6 + 12 + 1 + 3 = 22 ] Therefore, the total amount Laura needed to pay for all the products is boxed{22}.